Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245204 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1688 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1688 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,9)(5,19)(6,11)(7,12)(8,14)(10,17)(15,28)(16,23)(18,27)(20,25)(21,26)(22,32)(24,31)(29,30);(4,5)(8,22)(9,19)(13,28)(14,20)(18,21)(23,29)(24,25)(26,27)(31,32);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,23)(10,22)(11,24)(12,25)(13,19)(17,30)(18,31)(21,32)(26,28)(27,29);(1,4,17,21)(2,9,10,26)(3,14,30,32)(5,6,18,7)(8,23,22,28)(11,27,12,19)(13,24,29,25)(15,31,16,20)Polycyclic presentation:
2^9|d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fi,e^c=eh,f^c=fi,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152419);